Confronting Reform
Jeremy Kilpatrick

American Mathematical Monthly, volume 104 (1997), 955-962.

Three times in this century, some members of the American mathematical community have attempted to reform school mathematics only to discover that others objected to the direction the reform appeared to be taking. In this paper, I sketch the issues at stake in the first two reform efforts and then turn to the third and current effort, giving particular attention to the critique offered by H. Wu [33]. The paper ends with some thoughts on the challenges of changing school mathematics.

A UNIFIED CURRICULUM.At the turn of the century, reformers at the University of Chicago High School and at several other Illinois schools were attempting to unify the secondary curriculum, principally by merging the year-long courses in algebra and geometry [25]. In 1903, E.H. Moore [15], retiring as president of the American Mathematical Society, gave a powerful impetus to the nascent reform efforts by devoting part of his presidential address to mathematics in secondary education. Moore called for "the unification of pure and applied mathematics" and "the correlation of the different subjects," to be accomplished by organizing algebra, geometry, and physics into a "thoroughly coherent four years' course."

Reaction to what was to become known as the "Chicago movement" was swift. Conservative mathematicians in the East, most prominently David Eugene Smith, although tolerant of the brash Midwesterners tinkering with new approaches, argued that the secondary classroom was a place for pure, not applied, mathematics. In particular, the mental disciplinary power of geometry, together with its aesthetic and cultural value, demanded that it be kept in a separate course.

By the time the final report of the MAA's National Committee on Mathematical Requirements [13] appeared in 1923, much of the support for a unified curriculum had shifted to Grades 7 to 9 and away from Grades 10 to 12. The report's authors wanted all students, many of whom were dropping out of school by the end of Grade 9, to have a broad view of mathematics and consequently proposed some integrated courses for the junior high school. They also suggested ways in which the curriculum of Grades 10 to 12 might be reorganized to connect algebra with geometry and to include some work in statistics and even calculus. They acknowledged, however, that although experimental unified courses were being developed, few high schools were adopting them. The movement to unify the mathematics curriculum was already fading away under attacks on mathematics as a required subject in secondary school and the growth of courses emphasizing the social uses of mathematics (primarily arithmetic). Today, the residue of the reform effort can be seen in the "general mathematics" course, the impoverished counterpart to first-year algebra.

A MODERN CURRICULUM.
The next wave of reform began to build in the 1950's, as university mathematicians and school mathematics teachers joined forces to attempt to bridge what they saw as the widening gap between school and collegiate mathematics. Concerned that he "explosive development of mathematics" [4, p. 1] was not being reflected in the school curriculum, that too few students were entering college prepared to study advanced mathematics, and that the nation risked a serious shortage of mathematically trained personnel, reformers mounted a variety of projects to improve the teaching of school mathematics by developing new curriculum materials and retraining teachers. These efforts became known as the "new math"--"a label not so much for a cohesive set of reform proposals and activities as for an era during which a variety of reforms were undertaken" [26, p. 413].

Many of the reforms, but not all, were marked by an emphasis on what were seen as unifying concepts of mathematics--set, relation, function, and the like--coupled with the abstract structures--groups, rings, fields, vector spaces--into which they are organized. "Because the university mathematicians who dominated the modern mathematics movement tended to be specialists in pure rather than applied mathematics, they saw pure mathematics, with an emphasis on set theory and axiomatics, not only as the content that was missing from the school curriculum but also as providing the framework around which to reorganize that curriculum" [26, p. 412].

Again, the reaction was swift. Morris Kline was the first and loudest voice, arguing that aspects of the reform efforts were "'wholly misguided,' 'sheer nonsense,' attempts to replace the 'fruitful and rich essence of mathematics' with sterile, peripheral, pedantic details'" [quoted in 6, p. 55]. In the position paper "On the Mathematics Curriculum of the High School" [16]. Kline and 64 other mathematicians offered a more measured critique--essentially arguing that anyone attempting reform needed to link school mathematics more closely to its history and to concrete applications and not to make it so abstract and formal that future nonmathematicians would be turned away. An important feature of the paper was that it offered "fundamental principles and practical guidelines." E.G. Begle [3], director of the School Mathematics Study Group, the largest and most prominent of the new math curriculum reform projects, expressed delight with the new guidelines, claiming that most were reflected in the new textbooks, and then gently chided the authors for failing to distinguish among the different projects and their suggestions for curriculum improvement, thus effectively rejecting them all.

Once most of the new math projects had ended, Kline fired the last shot. In Why Johnny Can't Add: The Failure of the New Math [12], published in 1973, he reiterated and elaborated his opposition to the reform. Although the book was marred by a sometimes flippant tone and a persistent unwillingness to make distinctions among reform efforts, Kline offered cogent thoughts on deductions, rigor, and the language of mathematics. (Despite the book's title, it dealt with the secondary curriculum and not the teaching of arithmetic. Kline once confided that his publisher insisted on the title.) He ended the book by arguing that the appropriate direction for any reform "should be diametrically opposite to that taken by the new mathematics" [12, p. 144], toward mathematics as an integral part of a liberal education, with connections to culture, history, science, and other subjects. He cited with approval Moore's [15] call to combine mathematics with science in high school and to reduce the artificial separation between its pure and applied sides. Thus, Moore, who had pushed the earlier reform effort, was cast in opposition to the second.

The residue of the new math era may be difficult to see in today's school mathematics, but it is there. The precalculus course, for example, is a direct descendant of the elementary functions and introductory analysis courses that appeared during that time. Some of the new math's terminology and notation has disappeared, but much survives. And various topics, such as inequalities, that the reforms introduced into school mathematics have remained. With respect to changes in the way mathematics has been taught, few of the reform proposals appear to have been extensively implemented. It is popular to declare that the new math was tried and that it failed. Studies of school practice at that time [8], [17], however, suggested that in most classroom the reforms were never really tried.

A STANDARDS-BASED CURRICULUM.Over the past decade or so, the reform impulse has heated up once more, this time led by professional organizations under the banner of raising expectations and providing mathematical literacy for all [19], [20], [21], [22]. The publication A Nation at Risk [18] set many of the terms of the discourse: low performance on international assessments threatened the nation's economic competitiveness; declining test scores nationally meant that rigorous and measurable standards were needed. Reformers of school mathematics have argued that changes in society and within mathematics itself necessitate a more demanding school mathematics curriculum. The goal is to develop students' mathematical power: "Truth and beauty, utility and application frame the study of mathematics like the muses of Greek theater. Together, they define mathematical power, the objective of mathematics education" [22, p. 43].

Much of the leadership in promoting reform has come from the National Council of Teachers of Mathematics (NCTM). This organization, founded in 1920 to help preserve the place of mathematics in the secondary curriculum, supported but did not lead the new math reform efforts. Two decades ago, however, it began to play a more active role as a national voice for teachers, in part as a response to the widely perceived failure to change school mathematics during the new math era and in part to counter the ensuing "back to basics" backlash of the mid-1970s [14, pp. 22-25]. In its first, and most influential, reform document [19], the NCTM took the term standard from the rhetoric of raised expectations and accountability for results and made it a statement for judging the quality of school mathematics and for providing "an informed vision of the future" [14, p. 36].

The language of "mathematical power" represents an attempt to provide a vision of "what it means to be mathematically literate both in a world that relies on calculators and computers to carry out mathematical procedures and in a world where mathematics is rapidly growing and is extensively being applied in diverse fields" [19, p. 1]. The arguments given for reforming the school mathematics curriculum, instruction, and assessment rest on the contention that because "all industrialized countries have experienced a shift from an industrial to an information society," the mathematics that students need to know in order to be "self-fulfilled, productive citizens in the next century" [19, p. 3] has also changed. The changes in society have demanded that school change as well. Although previous reform efforts had their effects, virtually all observers of U.S. school mathematics classrooms have come away convinced that change is needed.

A large part of the standards-based reform is built on the view that mathematics itself has become more computational and less formal. "In recent years, a reaction against formalism has been growing. In recent mathematical research, there is a turn toward the concrete and the applicable. In texts and treatises, there is more respect for examples, less strictness in formal exposition" [5, p. 344]. Even before the recent controversy over "the death of proof" [1], [7], [10], so-called informal geometry courses, minus proof, were being introduced into high schools as state legislatures and school districts mandated "geometry for all." For some high school teachers, the call for "decreased attention" to "Euclidean geometry as a complete axiomatic system" and "two-column proofs" [19, p. 127] has been interpreted as permission to do away with proof altogether, for everyone [14, p. 118]. (For an eloquent defense of proof in high school geometry, together with a proposed year-long syllabus, see [32].) And clearly, the availability of computer software and graphing calculators has made it easier than ever before to visualize relationships and test numerous cases of a generalization before or in place of providing deductive justifications.

This time around, the negative reaction to reform proposals and activities has been slow to come. The lag may be because endorsements were sought and received for the proposals from all parts of the mathematical community. Or it may be because the proposals were framed in rather general terms, with textbooks and other materials from reform projects appearing only in the last few years. Opposition to the reform, however, has been building--much of it on the Web--for some time, and now we have an article in print by H. Wu [33], one of the most outspoken of the critics.

WU'S CRITIQUE.Like many before him, Wu is concerned with making distinctions among reform proposals or between the proposals and the various activities carried out and materials developed in the name of reform. He conflates what he terms the K-12 mathematics education reform promoted by the NCTM [19], [20], [21] and the calculus reform stimulated largely by work of the MAA and the National Academy of Sciences [27]. Although these two efforts share many common features, they have rather different agendas--with one attempting to lay out a broad framework for the mathematics all American schoolchildren need to know and be able to do and the other targeting a college course seen as unsatisfactory and out-of-date.

Wu's scattershot approach to the K-12 reform relies heavily for its force on unsubstantiated claims and random anecdotes. Contending that "the reform must be judged by its performance and not by its rhetoric" [33, p. 947], he offers no documentation of performance whatsoever. Instead, he often uses the rhetoric of the NCTM standards documents and the content of textbooks purporting to follow the standards to support his assertions that performance must be bad.

He also hits some inappropriate targets. For example, he castigates two high school teachers [24] writing in the Mathematics Teacher for their attempt to help students see the functions involved in a trigonometric identity before establishing its validity. Quite apart from whether every article in an official journal of an organization promoting reform must reflect reform views, one can reasonably ask whether seeing the graphs of these functions alone might not help students understand the identity. And how can Wu be so certain that teachers who are having students use graphing calculators are neglecting proof just because it is not mentioned in the article?

A second example is Wu's [33, p. 949] disapproval of textbooks that neglect "basic formulas." He cites the precalculus textbook produced by teachers at the North Carolina School of Science and Mathematics [2], a book whose origins were independent of, although ultimately in harmony with, the NCTM reform. The North Carolina approach relied much more than anything NCTM has proposed on the view that every topic ought to be introduced with an application [11, p. 154]. Wu notes that in its discussion of radian measure [2, pp. 209-210], the North Carolina textbook fails to give the formula relating degrees and radians but instead leaves it to the exercises. The issue here, as any textbook author will recognize, is the tension between textbook as archive and textbook as tool for learning. Once a formula is put into a text for memorization and subsequent reference, there is little point in asking the reader to find it. The omission of a formula from a textbook, reform or otherwise, should not be interpreted to imply that teachers will slight it in the classroom. Is there no value in having students develop a formula on their own? Must it always be given in the textbook? Wu is on much firmer ground elsewhere when he discusses problems associated with the use of open-ended problems [31]. The examples he gives clearly show that the authors of such problems have not always thought through the mathematics needed to solve them and the reasons for having students work on them.

A final example of Wu's indiscriminate approach is his criticism of the NCTM standards [19] for presumably arguing that information about the nature and existence of polynomial roots should be withheld from most high school students [33, p. 948]. The reference is to a discussion of how the solution of polynomial equations might be "differentiated in both depth and the level of formalism" [19, p. 152] by treating it at any of five levels. Nowhere does the document say that as a rule students should not learn about polynomial roots. What the discussion attempts is an illustration of how teachers might begin with exploration rather than abstraction, depending on the students' knowledge and experience. Wu sees this approach as advocacy of a utilitarian curriculum that never reaches "mathematical closure" (i.e., formal proof). He is right that the arguments for a standards-based curriculum are largely utilitarian, but he wrongly attributes the apparent decline in attention to proof to the standards movement alone, and he makes the same error some teachers do in interpreting a call for decreased attention to certain proof practices as sanctioning the complete elimination of formal proof itself.

The conception of the learner underlying the NCTM standards has usually been characterized as constructivism [14, p. 113], a term that has almost lost its meaning in American mathematics education. It began as the epistemological position, associated with Jean Piaget, that a learner both incorporates novel experience into existing mental structures (assimilation) and reorganizes those structures to handle more problematic experience (accommodation). Later interpreters stressed the accommodation aspect, arguing that learners actively construct knowledge rather than receiving it passively from the environment. The most radical view, which has become popular in some quarters of American mathematics education but almost nowhere else, is that the learner is an informationally closed system that cannot know an independent, pre-existing external world [29], [30].

As a theory of knowledge acquisition, constructivism says nothing about how teaching or instruction should proceed. In recent years, however, practices that encourage students to become active learners by conducting investigations, working in groups, and handling concrete objects have come to be characterized as "constructivist teaching." Only if educators all the way back to Plato--including Comenius, Pestalozzi, Herbart, Froebel, Dewey, and Montessori--were to be considered constructivists would such practices uniquely define constructivist teaching. Overblown claims have been made that radical constructivism has brought "a new revolution in mathematics education of a magnitude no less than the modern mathematics movement of the 1960s" [28, p. 720] and has provided the epistemology underlying the current reform, but there is no real evidence for these assertions. What is clear, however, is that the reform documents advocate some pedagogical practices that 50 years ago might have been labeled "progressive," but that today are termed "constructivist."

In his critique, Wu condemns "the new pedagogy, which relies heavily on constructivistic instructional strategies, such as cooperative learning [a form of group work not directly connected to the standards-based reform] and the discovery method [apparently any instructional approach in which students engage in inquiry]" [33, p. 949]. He claims that "too much of this is happening in the reform classrooms to the detriment of good education" [33, p. 950]. It is impossible to know, in the absence of any data, what constitutes heavy reliance or "too much."

Wu cites an article [9] that he says laments the absence of large-scale studies supporting "unrestricted" (his term) applications of cooperative learning. Although the article does not actually make that assertion, a more critical point is that research, no matter how large in scale, can never justify the indiscriminate use of a teaching method, traditional or otherwise. Methods are always interpreted and used in different ways by different teachers for teaching different topics to different students. A new method can no more be shown universally superior than can traditional methods, whatever they might be.

The most constructive part of Wu's critique, by far, is the final section in which he urges mathematicians to become more involved in mathematics education, contributing ideas to the revision of the NCTM standards documents, helping to improve the training of prospective teachers, and participating directly in curriculum change. Many mathematicians have already been involved in the current reforms for some time, but greater participation--encouraging or critical--can only be beneficial.

To progress as a field in how we deal with efforts to improve school mathematics, however, we need not only greater participation but also a higher level of discourse about those efforts. Critiques need to be based on substantive analyses that are grounded in evidence. They should consist of more than capricious assertions and bleak prophesies. We need to move from anecdote to analysis, from evisceration to evidence, from diatribe to dialogue.

SUPPORTING ONE ANOTHER.
Over the past century, the American mathematical community has become one of the most cohesive academic communities in the world. Few disciplines anywhere, for example, have organization such as the Conference Board of the Mathematical Sciences or the Mathematical Sciences Education Board to unite all elements of the community, from elementary school teachers to applied mathematicians working in industry. At conferences bringing together selected teachers, professors, and other scholars from across the continent to discuss education problems, those concerned with mathematics are almost invariably the first to coalesce, with many already well acquainted with one another. Since the American mathematical research community emerged over a century ago [23], American mathematics education has benefited from a virtually continual stream of support from prominent research mathematicians who have taken an interest in education, been willing to speak out for it publicly, and helped work for educational change.

Despite the cohesiveness, however, strains appear from time to time when the school mathematics curriculum is under scrutiny. These strains can be expressed as polar opposites, although of course there is always a spectrum of opinion in between. Some favor pure mathematics; others applied. Some want mathematics taught as they learned it; others want a different approach. Some are concerned primarily with developing the next generation of mathematicians; others are concerned primarily with mathematical literacy for all. For some, the deductive side of mathematics is what counts; others prefer the empirical, fallibilist, culturally determined side.

Whenever the mathematics taught in schools seems especially removed from mathematics seen as a scientific discipline and human enterprise, the strains can become especially great. That has happened at the beginning of the century, at mid-century, and now as the century draws to a close. The tension that these disagreements entail should remind us that if we did not share so much in common, we would not have such good grounds on which to disagree and to work toward a resolution.

Change in education is notoriously complex, difficult, and unpredictable. Reform movements in mathematics education turn out neither as advocates hope nor as detractors fear. But these movements can energize those teachers who want, as Begle once put it, to teach better mathematics and to teach mathematics better. As teachers struggle to improve their practice, a reform vision can provide needed direction, and membership in a mathematical community can provide needed support.

ACKNOWLEDGEMENTS.
I am grateful to Alan Schoenfeld, Guershon Harel, and George Stanic for comments on an earlier version of the paper and to Mark Freitag for observations on the treatment of proof in recent textbooks